Some grammar case studies

Imports and useful functions

try:
    # in colab
    import google.colab
    print('In colab, downloading LOTlib3')
    !git clone https://github.com/thelogicalgrammar/LOTlib3
    !pip install ColabTurtlePlus
    !pip install cairosvg
except:
    # not in colab
    print('Not in colab!')

from numpy.random import choice
import numpy as np
import matplotlib.pyplot as plt
import re
from ColabTurtlePlus import Turtle as t
import ColabTurtlePlus as ctp
import IPython

import cairosvg
import io
from PIL import Image, ImageFilter
from scipy.signal import sawtooth
from pprint import pprint

Not in colab!
Put clearscreen() as the first line in a cell (after the import command) to re-run turtle commands in the cell

from LOTlib3.Miscellaneous import q, random, logsumexp
from LOTlib3.Grammar import Grammar
from LOTlib3.DataAndObjects import FunctionData, Obj
from LOTlib3.Hypotheses.LOTHypothesis import LOTHypothesis
from LOTlib3.Eval import primitive
from LOTlib3.Miscellaneous import qq
from LOTlib3.TopN import TopN
from LOTlib3.Samplers.MetropolisHastings import MetropolisHastingsSampler

def normalize(arr):
    # Transform the array so that it sums to 1
    return arr / np.sum(arr)


def apply_rule(match, grammar):
    """
    Accepts both PCFGs (where the keys of grammar
    are lists of tuples (unnormalized prob, right hand side)
    and CFGs (where the keys are lists of right hand side strings.
    """
    match_txt = grammar[match.group(0)]
    try:
        # if PCFG
        probs, subs = zip(*match_txt)
    except ValueError:
        # if CFG
        subs = match_txt
        probs = [1]*len(subs)
    sub = choice(subs, p=normalize(probs))
    return sub


def complete(sentence, grammar):
    # if there are still '<' that means there are still nonterminals
    while '<' in sentence:
        # substitute the first nonterminal applying 
        # one of the rules in the grammar
        sentence = re.sub(
            # Greedy search of an expression between
            # angle brackets.
            '<(.*?)>', 
            lambda match: apply_rule(match, grammar), 
            sentence
        )
    return sentence

Binary string grammar

grammar = Grammar(start='STR')

Shapes

Some shape-specific functions

def get_svg(screen=None):
        """Shows the SVG code for the image to the screen."""
        if screen is None:
            screen = t.Screen()
        header = ("""<svg width="{w}" height="{h}" viewBox="0 0 {w} {h}" xmlns="http://www.w3.org/2000/svg">\n""").format(
            w=screen.window_size[0],
            h=screen.window_size[1]) 
        header += ("""<rect width="100%" height="100%" style="fill:{fillcolor};stroke:{kolor};stroke-width:1" />\n""").format(
            fillcolor=screen.background_color,
            kolor=screen.border_color)
        # lines = screen._svg_drawlines_string.replace("/>","/>\n")
        image = screen._generateSvgLines().replace("/>","/>\n") 
        output = header + image + "</svg>"
        return output

def blur_img(img, p_noise=0.):
    # whether to switch pixel or not
    # noise should be maximal when p_noise is 1
    p_noise = p_noise/2
    salt_and_pepper = np.where(
        np.random.choice(
            [0,1],
            size=img.shape,
            p=[1-p_noise, p_noise]
        ),
        img,
        1-img
    )
    return salt_and_pepper

def svg_to_img(svg, p_noise=0):
    """
    Load an SVG file and return image in Numpy array.
    Modified version of function from:
    https://stackoverflow.com/questions/55440483/
    how-can-i-optimize-the-palette-image-size-with-pil
    /55442505#55442505
    """
    # Make memory buffer
    mem = io.BytesIO()
    # Convert SVG to PNG in memory
    cairosvg.svg2png(
        bytestring=svg, 
        write_to=mem,
    )
    # Convert PNG to Pillow object
    img = np.array(Image.open(mem))
    # Remove greys and normalize in [0,1]
    # and take only first value of 3 RGB 
    # (They're all the same since original image
    # is in shades of grey)
    img = np.round(img[:,:,0]/img.max())
    return blur_img(
        np.array(img), 
        p_noise
    )

A grammar of shapes

Based on these movements, we can define a grammar that produces series of instructions and just use the meanings defined by the Turtle library:

(6)\[\begin{align} \langle\text{START}\rangle &\rightarrow \text{chain}(\langle\text{DRAWING}\rangle) \\ \langle\text{DRAWING}\rangle &\rightarrow \langle\text{MOVEMENT}\rangle,\langle\text{DRAWING}\rangle \\ &\rightarrow \epsilon \\ \langle\text{MOVEMENT}\rangle &\rightarrow \text{forward}(\langle\text{DISTANCE}\rangle) \\ &\rightarrow \text{backward}(\langle\text{DISTANCE}\rangle) \\ &\rightarrow \text{right}(\langle\text{ANGLE}\rangle) \\ &\rightarrow \text{left}(\langle\text{ANGLE}\rangle) \\ &\rightarrow \text{circle}(\langle\text{DISTANCE}\rangle, \langle\text{ANGLE}\rangle) \\ &\rightarrow \text{goto}(0,0) \\ \langle\text{DISTANCE}\rangle &\rightarrow 50 \mid 100 \mid 120 \\ \langle\text{ANGLE}\rangle &\rightarrow 30 \mid 60 \mid 90 \mid 170 \end{align}\]

grammar = {
    '<start>': [(1,'chain(<drawing>)')],
    # always start with a drawing
    '<drawing>': [
        # a drawing can include a movement 
        # follows by a drawing (recursive!)
        (8, '<movement>,<drawing>'),
        # at some point the drawing ends
        (1, ''),
    ],
    # a movement is basically a single instruction
    # so a drawing consists of a series of movements
    # Movements have two types of parameters,
    # namely: angles and distances
    '<movement>': [
        (1, 'forward_(<distance>)'),
        (1, 'backward_(<distance>)'),
        (1, 'right_(<angle>)'),
        (1, 'left_(<angle>)'),
        (1, 'circle_(<distance>,<angle>)'),
        (1, 'goto_(0,0)')
    ],
    # hand-specify some distances
    '<distance>': [
        (1, '10'),
        (1, '20'),
        (1, '50'),
    ],
    # hand-specify some angles
    '<angle>': [
        (1, '30'),
        (1, '60'),
        (1, '90'),
        (1, '170')
    ]
}

Turtle semantics

Instead of defining meanings ourselves, we can rely on an already existing library. For instance, consider Python’s ‘turtle’ library. The idea is very simple: Imagine a little turtle that moves around the screen according to certain instructions, and as it moves around it traces a line. Instructions are things like ‘move forward’, ‘go around some of a circle’, etc.

Each sentence in our grammar gets interpreted by the turtle library into a shape.

Here’s an example to show you the syntax:

t.clearscreen()
t.initializeTurtle()
t.hideturtle()
t.color('black')
t.speed(13)
t.forward(100); t.circle(50,90); t.forward(100)

t.clearscreen()
t.initializeTurtle()
t.hideturtle()
t.color('black')
t.speed(13)
t.forward(10); t.circle(20,60); t.right(90); t.forward(100); t.goto(0,0)

Generating from a grammar and interpreting

@primitive
def forward_(x): return lambda : t.forward(x)

@primitive
def backward_(x): return lambda : t.backward(x)

@primitive
def right_(x): return lambda : t.right(x)

@primitive
def left_(x): return lambda : t.left(x)

@primitive
def circle_(x,y): return lambda : t.circle(x,y)

@primitive
def goto_(x,y): return lambda : t.goto(x,y)

@primitive
def chain(*args, speed=0):
    # args is a list of functions
    t.initializeTurtle(window=(100,100))
    t.hideturtle()
    t.color('black')
    t.speed(speed)
    for f in args:
        f()
    screen = t.Screen()
    return get_svg(screen)

Try to run this multiple times and note that sometimes the result is just empty or a straight line, and sometimes it is some interesting complex shapes:

n_imgs = 3

imgs = []
for i in range(n_imgs):
    # define the 'central' part of the instructions
    instructions = complete('<start>', grammar)

    print(instructions)
    # add some needed scaffolding around the 
    # sentence produced by the grammar
    svg_representation = eval(instructions)
    imgs.append(
        svg_to_img(
            svg_representation,
        )
    )

fig, axes = plt.subplots(
    1,
    len(imgs), 
    sharey=True, 
    sharex=True
)

for ax, im in zip(axes, imgs):
    ax.imshow(im)

axes[0].set_xticks([])
axes[0].set_yticks([])
fig.savefig('imgs/shapes_examples.png')

chain(circle_(20,30),circle_(50,30),left_(170),forward_(10),forward_(20),backward_(20),goto_(0,0),circle_(50,30),goto_(0,0),circle_(50,170),backward_(10),forward_(20),right_(30),right_(30),backward_(50),circle_(20,90),circle_(10,60),circle_(10,30),right_(90),left_(90),circle_(50,30),backward_(10),right_(90),left_(60),backward_(50),left_(90),right_(90),backward_(50),)
chain(left_(60),forward_(10),backward_(50),left_(30),circle_(20,170),backward_(50),goto_(0,0),goto_(0,0),goto_(0,0),forward_(10),left_(30),backward_(20),right_(30),circle_(10,90),right_(30),right_(170),forward_(20),circle_(20,170),left_(90),backward_(10),left_(90),left_(60),circle_(20,60),backward_(10),left_(90),right_(90),circle_(10,90),)
chain(goto_(0,0),left_(170),backward_(50),right_(60),goto_(0,0),right_(60),circle_(10,60),circle_(20,90),circle_(20,170),forward_(20),right_(60),forward_(50),forward_(10),left_(30),forward_(10),backward_(10),right_(90),left_(60),left_(90),forward_(50),circle_(10,90),left_(60),forward_(50),left_(90),left_(30),forward_(10),forward_(50),right_(90),circle_(10,60),backward_(10),)

The point here is: each sentence in our grammar is a shape.

If we take the LoT hypothesis seriously, we can then think that shape concepts (squares, triangles, etc.) are encoded at the cognitive level as sentences in the geometric fragment of the LoT.

Practical note: Sometimes we do not want to see the full animation, but rather just the final product. In that case, we can set speed at 0 and then call done, as follows:

t.clearscreen()
t.initializeTurtle()
t.hideturtle()
t.color('black')
t.speed(0)
for i in range(20): {instructions}
t.done()

Going from a shape to a noisy observation

n_shapes=3
degrees_noise = [0, 0.4, 0.7, 1.0]

images = []
for i in range(n_shapes):
    instructions1 = complete('<start>', grammar)
    print(instructions1, "\n")
    svg_representation = eval(instructions1) 
    for d in degrees_noise:
        img = svg_to_img(
            svg_representation,
            d
        )
        images.append(img)

fig, axes = plt.subplots(
    n_shapes,
    len(degrees_noise), 
    sharex=True, 
    sharey=True
)

for ax in axes.flatten():
    ax.set_xticks([])
    ax.set_yticks([])

for ax, im in zip(axes.flatten(), images):
    ax.imshow(im)

fig.tight_layout()
fig.savefig('imgs/noisy_vs_sharp.png', dpi=300)

chain(circle_(50,90),left_(60),left_(30),backward_(20),right_(90),right_(170),circle_(50,60),backward_(10),circle_(50,60),right_(90),goto_(0,0),forward_(20),circle_(50,60),backward_(10),goto_(0,0),backward_(50),) 

chain(goto_(0,0),forward_(10),right_(30),right_(170),circle_(50,30),circle_(50,90),left_(170),goto_(0,0),goto_(0,0),left_(170),right_(170),) 

chain(right_(90),backward_(20),goto_(0,0),right_(170),circle_(50,170),circle_(20,90),right_(170),goto_(0,0),circle_(10,30),left_(90),goto_(0,0),left_(30),forward_(50),right_(90),right_(170),goto_(0,0),backward_(10),forward_(20),forward_(50),right_(170),) 

Learning a shape from an image

##### define a hypothesis
class MyHypothesis(LOTHypothesis):
    
    def __init__(self, **kwargs):
        LOTHypothesis.__init__(
            self, 
            grammar=lot_grammar, 
            display="lambda: %s", 
            **kwargs
        )

    def compute_single_likelihood(self, datum):
        # eval expression of current hypothesis 
        # to get an svg image, then get pixels
        img_clear = svg_to_img(
            self(),
            # level of noise
            p_noise=0
        )
        
        # total logprob
        total_loglik = np.where(
            img_clear==datum.output,
            np.log(1-datum.noise_level),
            np.log(datum.noise_level)
        ).sum()
        
        return total_loglik
        
    def display():
        svg_representation = self()
        img_clear = svg_to_img(
            svg_representation,
            # degree of blurriness
            0
        )
        plt.imshow(img_clear)

print("Defined a hypothesis class")

Defined a hypothesis class

##### define a LOTlib3 grammar
lot_grammar = Grammar(start='<start>')
for nonterminal, substitutions in grammar.items():
    for weight, subt in substitutions:

        to_substitute = []
        def capture_and_replace(match):
            to_substitute.append(match.group(0))
            return '%s'
        # replace nonterminals with %s
        subt_replaced = re.sub(
            r'<[a-z]*>', 
            capture_and_replace, 
            subt
        )
        # add a rule to the grammar
        print(subt_replaced)
        lot_grammar.add_rule(
            nonterminal, 
            subt_replaced, 
            to_substitute if len(to_substitute)>0 else None, 
            weight
        )

print("Defined the grammar!")

chain(%s)
%s,%s

forward_(%s)
backward_(%s)
right_(%s)
left_(%s)
circle_(%s,%s)
goto_(0,0)
10
20
50
30
60
90
170
Defined the grammar!

# define the 'central' part of the instructions
instructions = complete('<start>', grammar)

noise_level = 0.1
print(instructions)
# add some needed scaffolding around the 
# sentence produced by the grammar
svg_representation = eval(instructions)
img_noisy = svg_to_img(
    svg_representation,
    p_noise=noise_level
)

plt.imshow(img_noisy)

chain(circle_(50,90),backward_(10),right_(90),right_(170),right_(30),forward_(10),right_(90),left_(170),goto_(0,0),circle_(20,60),right_(30),goto_(0,0),goto_(0,0),goto_(0,0),backward_(20),forward_(20),goto_(0,0),circle_(10,60),circle_(20,60),circle_(50,60),left_(170),goto_(0,0),right_(60),left_(30),)

<matplotlib.image.AxesImage object at 0x7f33c55fa650>

We have defined an interpreted grammar for ideal, abstract shapes. Imagine now the following situations. We see some image and we are trying to learn what shape (in our shape grammar) it is. The problem is that the images are blurry, so we are not quite sure: Each image could have been generated by different shapes, to varying extents!

h = MyHypothesis()

##### Define Data
data = [ 
    FunctionData(
        input=[], 
        output=img_noisy, 
        noise_level=noise_level
    )
]

# store the top N
tn = TopN(
    N=1000
) 

print("Defined the data")

##### Run inference
h = MyHypothesis()

print("Running inference...")

Defined the data
Running inference...

for i, h in enumerate(MetropolisHastingsSampler(h, data, steps=40000)):
    tn.add(h)
    if i%100==0:
        print(i, end=', ')

0, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200, 1300, 1400, 1500, 1600, 1700, 1800, 1900, 2000, 2100, 2200, 2300, 2400, 2500, 2600, 2700, 2800, 2900, 3000, 3100, 3200, 3300, 3400, 3500, 3600, 3700, 3800, 3900, 4000, 4100, 4200, 4300, 4400, 4500, 4600, 4700, 4800, 4900, 5000, 5100, 5200, 5300, 5400, 5500, 5600, 5700, 5800, 5900, 6000, 6100, 6200, 6300, 6400, 6500, 6600, 6700, 6800, 6900, 7000, 7100, 7200, 7300, 7400, 7500, 7600, 7700, 7800, 7900, 8000, 8100, 8200, 8300, 8400, 8500, 8600, 8700, 8800, 8900, 9000, 9100, 9200, 9300, 9400, 9500, 9600, 9700, 9800, 9900, 10000, 10100, 10200, 10300, 10400, 10500, 10600, 10700, 10800, 10900, 11000, 11100, 11200, 11300, 11400, 11500, 11600, 11700, 11800, 11900, 12000, 12100, 12200, 12300, 12400, 12500, 12600, 12700, 12800, 12900, 13000, 13100, 13200, 13300, 13400, 13500, 13600, 13700, 13800, 13900, 14000, 14100, 14200, 14300, 14400, 14500, 14600, 14700, 14800, 14900, 15000, 15100, 15200, 15300, 15400, 15500, 15600, 15700, 15800, 15900, 16000, 16100, 16200, 16300, 16400, 16500, 16600, 16700, 16800, 16900, 17000, 17100, 17200, 17300, 17400, 17500, 17600, 17700, 17800, 17900, 18000, 18100, 18200, 18300, 18400, 18500, 18600, 18700, 18800, 18900, 19000, 19100, 19200, 19300, 19400, 19500, 19600, 19700, 19800, 19900, 20000, 20100, 20200, 20300, 20400, 20500, 20600, 20700, 20800, 20900, 21000, 21100, 21200, 21300, 21400, 21500, 21600, 21700, 21800, 21900, 22000, 22100, 22200, 22300, 22400, 22500, 22600, 22700, 22800, 22900, 23000, 23100, 23200, 23300, 23400, 23500, 23600, 23700, 23800, 23900, 24000, 24100, 24200, 24300, 24400, 24500, 24600, 24700, 24800, 24900, 25000, 25100, 25200, 25300, 25400, 25500, 25600, 25700, 25800, 25900, 26000, 26100, 26200, 26300, 26400, 26500, 26600, 26700, 26800, 26900, 27000, 27100, 27200, 27300, 27400, 27500, 27600, 27700, 27800, 27900, 28000, 28100, 28200, 28300, 28400, 28500, 28600, 28700, 28800, 28900, 29000, 29100, 29200, 29300, 29400, 29500, 29600, 29700, 29800, 29900, 30000, 30100, 30200, 30300, 30400, 30500, 30600, 30700, 30800, 30900, 31000, 31100, 31200, 31300, 31400, 31500, 31600, 31700, 31800, 31900, 32000, 32100, 32200, 32300, 32400, 32500, 32600, 32700, 32800, 32900, 33000, 33100, 33200, 33300, 33400, 33500, 33600, 33700, 33800, 33900, 34000, 34100, 34200, 34300, 34400, 34500, 34600, 34700, 34800, 34900, 35000, 35100, 35200, 35300, 35400, 35500, 35600, 35700, 35800, 35900, 36000, 36100, 36200, 36300, 36400, 36500, 36600, 36700, 36800, 36900, 37000, 37100, 37200, 37300, 37400, 37500, 37600, 37700, 37800, 37900, 38000, 38100, 38200, 38300, 38400, 38500, 38600, 38700, 38800, 38900, 39000, 39100, 39200, 39300, 39400, 39500, 39600, 39700, 39800, 39900, 

hs = tn.get_all(sorted=True)
z = logsumexp([h.posterior_score for h in hs])
pps_im = np.zeros(true_img.shape)
for h in hs:
    p = np.exp(h.posterior_score - z)
    print(np.round(p,3), end=', ')
    try:
        svg_rep = h()
        h_img = svg_to_img(svg_rep)
        print(np.unique(h_img), end=', ')
        print(h)
        pps_im = pps_im + h_img*p
    except:
        pass

0.0, 0.0, [0. 1.], lambda: chain(left_(90),left_(30),right_(170),backward_(50),backward_(20),)
0.0, [0. 1.], lambda: chain(goto_(0,0),left_(30),right_(170),backward_(50),backward_(20),)
0.0, [0. 1.], lambda: chain(goto_(0,0),left_(30),right_(170),backward_(50),forward_(50),)
0.0, [0. 1.], lambda: chain(goto_(0,0),left_(30),right_(170),right_(170),forward_(50),)
0.0, [0.], lambda: chain(goto_(0,0),left_(30),right_(170),right_(170),goto_(0,0),)
0.0, [0.], lambda: chain(goto_(0,0),left_(30),right_(170),right_(30),goto_(0,0),)
0.0, [0.], lambda: chain(goto_(0,0),left_(30),right_(170),left_(60),goto_(0,0),)
0.0, [0.], lambda: chain(goto_(0,0),left_(30),right_(170),goto_(0,0),goto_(0,0),)
0.0, [0.], lambda: chain(goto_(0,0),left_(30),right_(170),)
0.0, [0.], lambda: chain(goto_(0,0),left_(30),right_(90),)
0.0, [0.], lambda: chain(goto_(0,0),left_(60),right_(90),)
0.0, [0.], lambda: chain(left_(30),)
0.0, [0.], lambda: chain()
0.0, [0. 1.], lambda: chain(circle_(50,30),forward_(10),right_(90),circle_(50,170),right_(170),)
0.0, [0. 1.], lambda: chain(circle_(50,30),forward_(10),right_(90),circle_(50,170),left_(90),)
0.0, [0. 1.], lambda: chain(circle_(50,30),forward_(10),right_(90),circle_(50,170),forward_(50),)
0.0, [0. 1.], lambda: chain(circle_(50,30),forward_(10),right_(90),circle_(50,170),forward_(10),)
0.0, [0. 1.], lambda: chain(circle_(50,30),forward_(10),right_(90),circle_(50,170),forward_(20),)
0.0, [0. 1.], lambda: chain(circle_(50,30),forward_(10),right_(90),circle_(10,60),)
0.0, [0. 1.], lambda: chain(circle_(50,170),right_(60),right_(30),right_(170),forward_(10),backward_(50),)
0.0, [0. 1.], lambda: chain(circle_(50,170),right_(60),circle_(20,60),right_(170),forward_(10),)
0.0, [0. 1.], lambda: chain(circle_(50,90),right_(30),circle_(20,60),right_(170),forward_(20),)
0.0, [0. 1.], lambda: chain(circle_(50,170),right_(30),circle_(20,60),right_(170),forward_(20),)
0.0, [0. 1.], lambda: chain(circle_(50,170),right_(170),circle_(20,60),right_(170),forward_(20),)
0.0, [0. 1.], lambda: chain(circle_(50,170),right_(170),circle_(20,60),right_(170),forward_(10),)
0.0, [0. 1.], lambda: chain(circle_(50,90),right_(30),right_(170),left_(90),forward_(50),)
0.0, [0. 1.], lambda: chain(circle_(50,90),right_(30),right_(170),left_(30),forward_(50),)
0.0, [0. 1.], lambda: chain(circle_(50,90),right_(30),right_(170),left_(30),forward_(20),)
0.0, [0. 1.], lambda: chain(circle_(50,90),right_(30),right_(170),right_(170),forward_(20),)
0.0, [0. 1.], lambda: chain(circle_(50,170),right_(60),right_(30),right_(170),forward_(10),)
0.0, [0. 1.], lambda: chain(circle_(50,90),backward_(20),right_(60),forward_(20),left_(170),circle_(20,90),)
0.0, [0. 1.], lambda: chain(circle_(50,170),forward_(10),right_(90),circle_(10,60),)
0.0, [0. 1.], lambda: chain(circle_(50,90),right_(60),right_(170),right_(170),)
0.0, [0. 1.], lambda: chain(circle_(50,90),right_(30),right_(170),right_(170),)
0.0, [0. 1.], lambda: chain(circle_(50,90),right_(30),right_(170),left_(90),)
0.0, [0. 1.], lambda: chain(circle_(50,90),right_(60),right_(30),right_(170),)
0.0, [0. 1.], lambda: chain(circle_(50,170),forward_(10),right_(90),right_(90),)
0.0, [0. 1.], lambda: chain(circle_(50,170),forward_(20),right_(90),right_(90),)
0.0, [0. 1.], lambda: chain(circle_(50,90),forward_(20),right_(90),right_(90),)
0.0, [0. 1.], lambda: chain(circle_(50,90),right_(60),forward_(50),right_(30),)
0.0, [0. 1.], lambda: chain(circle_(50,90),right_(60),forward_(50),right_(170),)
0.0, [0. 1.], lambda: chain(circle_(50,170),forward_(10),right_(90),right_(170),)
0.0, [0. 1.], lambda: chain(circle_(50,90),forward_(20),forward_(50),right_(90),)
0.0, [0. 1.], lambda: chain(circle_(50,90),forward_(20),forward_(50),right_(30),)
0.0, [0. 1.], lambda: chain(circle_(50,90),backward_(20),right_(60),right_(30),left_(90),)
0.0, [0. 1.], lambda: chain(circle_(50,170),right_(170),right_(170),)
0.0, [0. 1.], lambda: chain(circle_(50,170),right_(170),right_(60),)
0.0, [0. 1.], lambda: chain(circle_(50,170),forward_(20),right_(60),)
0.0, [0. 1.], lambda: chain(circle_(50,170),right_(30),backward_(10),)
0.0, [0. 1.], lambda: chain(circle_(50,170),backward_(10),backward_(10),)
0.0, [0. 1.], lambda: chain(circle_(50,90),backward_(20),left_(60),)
0.0, [0. 1.], lambda: chain(circle_(50,90),backward_(20),left_(90),)
0.0, [0. 1.], lambda: chain(circle_(50,90),backward_(20),left_(30),)
0.0, [0. 1.], lambda: chain(circle_(50,90),backward_(20),right_(60),)
0.0, [0. 1.], lambda: chain(circle_(50,90),backward_(20),right_(90),)
0.0, [0. 1.], lambda: chain(circle_(50,90),backward_(20),right_(30),)
0.0, [0. 1.], lambda: chain(circle_(50,90),backward_(10),backward_(10),)
0.0, [0. 1.], lambda: chain(circle_(50,90),backward_(20),backward_(10),)
0.0, [0. 1.], lambda: chain(circle_(50,90),backward_(10),backward_(20),)
0.0, [0. 1.], lambda: chain(circle_(50,90),backward_(20),forward_(20),)
0.0, [0. 1.], lambda: chain(circle_(50,90),backward_(20),forward_(10),)
0.0, [0. 1.], lambda: chain(circle_(50,90),backward_(20),forward_(50),)
0.0, [0. 1.], lambda: chain(circle_(50,90),backward_(10),)
0.0, [0. 1.], lambda: chain(circle_(50,90),backward_(20),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),right_(60),left_(60),backward_(10),backward_(10),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),left_(170),left_(30),goto_(0,0),left_(90),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),left_(170),left_(90),goto_(0,0),left_(90),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),right_(90),left_(60),goto_(0,0),goto_(0,0),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),right_(30),left_(60),goto_(0,0),goto_(0,0),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),right_(30),right_(170),goto_(0,0),goto_(0,0),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),left_(30),right_(170),goto_(0,0),goto_(0,0),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),left_(90),right_(170),goto_(0,0),goto_(0,0),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),left_(60),right_(170),goto_(0,0),goto_(0,0),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),left_(170),right_(170),goto_(0,0),goto_(0,0),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),left_(170),left_(30),goto_(0,0),goto_(0,0),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),left_(90),goto_(0,0),left_(90),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),left_(170),goto_(0,0),left_(90),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),backward_(20),left_(60),goto_(0,0),goto_(0,0),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),right_(30),goto_(0,0),goto_(0,0),goto_(0,0),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),backward_(20),left_(90),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),backward_(20),left_(170),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),backward_(20),left_(60),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),backward_(10),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),left_(90),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),left_(60),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),right_(60),goto_(0,0),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),right_(30),goto_(0,0),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),goto_(0,0),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),backward_(20),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),right_(60),left_(60),goto_(0,0),backward_(10),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),right_(60),left_(60),goto_(0,0),backward_(20),left_(60),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),right_(60),left_(60),goto_(0,0),backward_(20),left_(170),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),right_(60),left_(60),goto_(0,0),backward_(20),left_(90),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),right_(60),left_(60),goto_(0,0),backward_(20),left_(30),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),right_(60),left_(60),goto_(0,0),backward_(20),right_(60),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),right_(60),left_(60),goto_(0,0),backward_(20),right_(170),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),right_(60),left_(60),goto_(0,0),backward_(20),right_(30),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),right_(60),left_(60),goto_(0,0),backward_(20),right_(90),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),right_(60),left_(60),goto_(0,0),backward_(20),forward_(10),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),right_(60),left_(60),goto_(0,0),backward_(20),forward_(20),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),right_(60),left_(60),backward_(20),backward_(20),goto_(0,0),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),right_(60),left_(60),goto_(0,0),backward_(10),backward_(10),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),right_(60),left_(60),backward_(10),goto_(0,0),backward_(20),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),right_(60),left_(60),backward_(20),goto_(0,0),backward_(20),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),right_(60),left_(60),goto_(0,0),backward_(20),goto_(0,0),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),right_(60),left_(60),goto_(0,0),goto_(0,0),backward_(20),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),backward_(20),right_(30),right_(170),goto_(0,0),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),backward_(20),left_(60),right_(170),goto_(0,0),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),backward_(20),left_(60),right_(30),goto_(0,0),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),backward_(20),left_(30),right_(30),goto_(0,0),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),backward_(20),left_(170),right_(170),goto_(0,0),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),backward_(20),left_(170),right_(90),goto_(0,0),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),backward_(20),right_(170),right_(90),goto_(0,0),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),backward_(10),goto_(0,0),backward_(20),goto_(0,0),right_(30),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),backward_(10),goto_(0,0),backward_(20),goto_(0,0),right_(170),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),right_(60),left_(60),backward_(20),backward_(20),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),backward_(20),backward_(10),backward_(10),right_(90),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),backward_(20),backward_(10),backward_(10),right_(170),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),backward_(20),backward_(10),backward_(10),forward_(20),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),backward_(20),backward_(10),backward_(10),forward_(10),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),backward_(20),goto_(0,0),right_(30),goto_(0,0),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),backward_(20),goto_(0,0),right_(60),goto_(0,0),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),backward_(20),goto_(0,0),right_(170),goto_(0,0),)
0.0, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),backward_(10),goto_(0,0),backward_(20),goto_(0,0),goto_(0,0),)
0.001, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),right_(60),left_(60),goto_(0,0),backward_(20),)
0.001, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),backward_(10),backward_(10),right_(30),)
0.001, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),backward_(10),backward_(10),right_(60),)
0.001, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),backward_(10),backward_(10),right_(90),)
0.001, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),backward_(10),backward_(10),right_(170),)
0.001, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),backward_(10),backward_(10),left_(30),)
0.001, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),backward_(10),backward_(10),left_(170),)
0.001, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),backward_(20),goto_(0,0),backward_(20),right_(30),)
0.001, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),backward_(20),goto_(0,0),backward_(20),right_(60),)
0.001, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),backward_(10),goto_(0,0),backward_(20),right_(60),)
0.001, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),backward_(10),goto_(0,0),backward_(20),right_(170),)
0.001, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),backward_(10),goto_(0,0),backward_(20),right_(90),)
0.001, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),backward_(10),goto_(0,0),backward_(20),right_(30),)
0.001, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),backward_(10),goto_(0,0),backward_(20),left_(170),)
0.001, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),backward_(10),backward_(10),forward_(10),)
0.001, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),backward_(10),backward_(10),forward_(20),)
0.001, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),backward_(20),backward_(20),forward_(10),goto_(0,0),)
0.001, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),backward_(20),backward_(20),forward_(20),goto_(0,0),)
0.001, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),backward_(10),goto_(0,0),backward_(20),forward_(20),)
0.001, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),backward_(10),goto_(0,0),backward_(20),forward_(10),)
0.002, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),backward_(20),goto_(0,0),right_(30),)
0.002, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),goto_(0,0),backward_(20),left_(170),)
0.002, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),goto_(0,0),backward_(20),right_(170),)
0.003, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),backward_(10),backward_(10),goto_(0,0),)
0.003, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),backward_(20),forward_(20),goto_(0,0),)
0.003, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),backward_(20),goto_(0,0),forward_(10),)
0.003, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),backward_(20),goto_(0,0),backward_(20),)
0.003, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),backward_(10),goto_(0,0),backward_(20),)
0.003, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),backward_(20),goto_(0,0),backward_(20),goto_(0,0),)
0.003, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),backward_(10),goto_(0,0),backward_(20),goto_(0,0),)
0.003, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),goto_(0,0),backward_(20),forward_(20),)
0.005, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),backward_(20),backward_(20),left_(60),)
0.005, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),backward_(20),backward_(20),left_(90),)
0.005, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),backward_(20),backward_(20),right_(170),)
0.007, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),backward_(20),backward_(20),forward_(20),)
0.007, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),backward_(20),backward_(20),forward_(10),)
0.007, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),backward_(10),backward_(10),backward_(20),)
0.009, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),backward_(20),goto_(0,0),goto_(0,0),)
0.009, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),goto_(0,0),backward_(20),goto_(0,0),)
0.016, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),backward_(20),backward_(20),circle_(10,60),)
0.016, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),backward_(20),backward_(20),circle_(10,30),)
0.016, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),backward_(20),right_(90),)
0.016, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),backward_(20),right_(30),)
0.016, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),backward_(20),right_(170),)
0.016, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),backward_(20),left_(60),)
0.021, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),backward_(10),backward_(10),)
0.021, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),backward_(20),backward_(20),goto_(0,0),)
0.021, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),backward_(20),forward_(10),)
0.021, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),backward_(10),goto_(0,0),backward_(20),)
0.021, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),backward_(20),goto_(0,0),backward_(20),)
0.063, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),backward_(20),goto_(0,0),)
0.063, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),goto_(0,0),backward_(20),)
0.142, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),backward_(20),backward_(20),)
0.425, [0. 1.], lambda: chain(circle_(50,90),goto_(0,0),forward_(20),goto_(0,0),backward_(20),)

true_img = svg_to_img(
    svg_representation,
    p_noise=0
)

best_str = tn.best()
best_img = svg_to_img(best_str())

fig, (ax_true, ax_observation, ax_best, ax_pps) = plt.subplots(
    1,4,
    sharex=True,
    sharey=True,
    figsize=(10,3)
)

kwargs = {
    'vmin': 0,
    'vmax': 1,
}

ax_true.imshow(true_img)
ax_true.set_xlabel('True img')
ax_observation.imshow(img_noisy, **kwargs)
ax_observation.set_xlabel('Noisy observation')
ax_best.imshow(best_img, **kwargs)
ax_best.set_xlabel((
    'Highest posterior img'
    # '\n'+best_str.__str__()'
))
ax_pps.imshow(pps_im, **kwargs)

ax_pps.set_xlabel('PPS img')
ax_true.set_xticks([])
ax_true.set_yticks([])

# fig.suptitle('True: '+instructions)
fig.tight_layout()
fig.savefig('imgs/inference.png')

fig, (ax_true, ax_observation, ax_best, ax_pps) = plt.subplots(
    1,4,
    sharex=True,
    sharey=True,
    figsize=(10,3)
)

kwargs = {
    'vmin': 0,
    'vmax': 1,
}

ax_true.imshow(true_img)
ax_true.set_xlabel('True img')
ax_observation.imshow(img_noisy, **kwargs)
ax_observation.set_xlabel('Noisy observation')
ax_best.imshow(best_img, **kwargs)
ax_best.set_xlabel('Highest posterior img\n'+best_str.__str__())
ax_pps.imshow(pps_im, **kwargs)

ax_pps.set_xlabel('PPS img')
ax_true.set_xticks([])
ax_true.set_yticks([])

fig.suptitle('True: '+instructions)
fig.tight_layout()

Things to try

Grammar

• How to make the shapes more round?

• How to make them more “spiky”?

Likelihood

• Can you think of other kinds of noise?

• How would they affect inference?

Sounds

Defining the grammar

Another fascinating domain of interest for cognitive science is that of sounds. Let’s think of what a simple grammar for sounds could be like. Let’s define a very simple grammar of music which can nonetheless encode many possible sounds:

(7)\[\begin{align} \langle\text{TUNE}\rangle &\rightarrow \langle\text{NOTE}\rangle \\ &\rightarrow \text{successive}(\langle\text{TUNE}\rangle,\langle\text{TUNE}\rangle) \\ &\rightarrow \text{melt}(\langle\text{TUNE}\rangle,\langle\text{TUNE}\rangle) \\ \langle\text{NOTE}\rangle &\rightarrow \text{defineNote}(\langle\text{FREQ}\rangle,\langle\text{TIME}\rangle) \\ \langle\text{TIME}\rangle &\rightarrow 0.1 \mid 0.2 \mid 0.3 \\ \langle\text{FREQ}\rangle &\rightarrow \text{A} \mid \text{Bb} \mid \text{B} \mid \text{C} \mid \text{Db} \mid \text{D} \mid \text{Eb} \mid \text{E} \mid \text{F} \mid \text{Gb} \mid \text{G} \mid \text{Ab} \end{align}\]

Define each note in an octave, so we can call them by name as described in the grammar:

freqs = {
    'A': 220,
    # 'Bb': 233.08,
    # 'B': 246.94,
    'C': 261.63,
    # 'Db': 277.18,
    'D': 293.66,
    # 'Eb': 311.13,
    'E': 329.63,
    # 'F': 349.23,
    # 'Gb': 369.99,
    'G': 392.00,
    # 'Ab': 415.30,
}

soundGrammar = {
    '<tune>': [
        (3, '<note>'),
        (1, 'successive(<tune>, <tune>)'),
        (1, 'melt(<tune>, <tune>)')
    ],
    '<note>': [
        (1, "defineNote('<freq>', <time>)")
    ],
    '<freq>': [(1,x) for x in freqs.keys()],
    '<time>': [
        (1, '0.3'), 
        (1, '0.6'), 
        (1, '1.'),
    ]
}

pprint(soundGrammar)

{'<freq>': [(1, 'A'), (1, 'C'), (1, 'D'), (1, 'E'), (1, 'G')],
 '<note>': [(1, "defineNote('<freq>', <time>)")],
 '<time>': [(1, '0.3'), (1, '0.6'), (1, '1.')],
 '<tune>': [(3, '<note>'),
            (1, 'successive(<tune>, <tune>)'),
            (1, 'melt(<tune>, <tune>)')]}

Semantics: From sentence to sounds

# define rate (don't worry about this!)
fs = 44100

We are going to need a way of defining notes given a frequency:

@primitive
def defineNote(frequency, seconds, fs=44100, harmonic_strength=0.1):

    if isinstance(frequency, str):
        frequency = freqs[frequency]
    # jitter frequency slightly to reduce strident interference
    # frequency = np.random.rand()*3 + frequency
    
    # seconds*sample_rate steps 
    # between 0 and seconds
    t = np.linspace(0, seconds, int(seconds * fs), False)

    # Harmonics
    harmonics = np.zeros_like(t)
    for i in range(2, 11):  # Adding the first 10 harmonics
        harmonics += (
            np.sin(frequency * i * t * 2 * np.pi)
            # sawtooth(2 * np.pi * i * frequency * t, 0.5)
            * np.exp(
                -i * harmonic_strength
            )
        )

    # Fundamental frequency
    note = np.sin(frequency * t * 2 * np.pi) + harmonics
    # note = sawtooth(2 * np.pi * frequency * t, 0.5)

    # # Create ADSR envelope
    # attack_time = 0.5
    # decay_time = 0.1
    # sustain_level = 1.
    # release_time = 0.1

    # total_frames = len(t)
    # attack_frames = int(total_frames * attack_time)
    # decay_frames = int(total_frames * decay_time)
    # sustain_frames = int(total_frames * (1.0 - attack_time - decay_time - release_time))
    # release_frames = total_frames - attack_frames - decay_frames - sustain_frames

    # envelope = np.concatenate([
    #     np.linspace(0, 1, attack_frames),  # Attack
    #     np.linspace(1, sustain_level, decay_frames),  # Decay
    #     np.full(sustain_frames, sustain_level),  # Sustain
    #     np.linspace(sustain_level, 0, release_frames)  # Release
    # ])

    # # Apply envelope
    # note *= envelope

    return note

We’ll also need a way of melting two notes, i.e. playing them together:

@primitive
def melt(note1, note2):
    # if they are of different lengths,
    # pad the shorter one with 0s
    diff = len(note1) - len(note2)
    if diff < 0:
        note1 = np.concatenate((
            note1,
            np.zeros(-diff, dtype=int)
        ))
    elif diff > 0:
        note2 = np.concatenate((
            note2,
            np.zeros(diff, dtype=int)
        ))

    combined = note1+note2
    audio = combined * (2**15 - 1) / np.max(np.abs(combined))
    audio = audio.astype(np.int16)
    
    return audio

And a way of appending one note to another one:

@primitive
def successive(note1, note2):
    return np.concatenate((note1, note2))

And finally collect all the primitive meanings we defined in a single dictionary, so that we can use it with interpret to get the meaning of each sentence produced by the grammar:

interpret = {
    **freqs,
    'defineNote': defineNote,
    'melt': melt,
    'successive': successive 
}

Creating music

Now we can play “music”:

simple_A = defineNote(220, 1, harmonic_strength=0.1)

IPython.display.Audio(
    # put note or notes here!
    simple_A, 
    rate=fs, 
    autoplay=True
)

plt.plot(simple_A)
plt.yticks([])
plt.savefig('imgs/simple_A.png')

Test the function:

melted = melt(
    defineNote(freqs['A'], 1),
    defineNote(freqs['D'], 2)
)

IPython.display.Audio(
    melted, 
    rate=fs, 
    autoplay=True
)

plt.plot(melted)
plt.yticks([])
plt.savefig('imgs/example_melt.png')

Run this until you get something long that looks interesting! As in the geometric example above, some productions are very simple while others are very messy and complicated.

tune = complete('<tune>', soundGrammar)
pprint(tune)

("successive(defineNote('A', 1.), successive(defineNote('E', 0.3), "
 "successive(defineNote('A', 0.3), melt(defineNote('D', 0.6), defineNote('E', "
 '1.)))))')

IPython.display.Audio(
    # put note or notes here!
    eval(tune, interpret), 
    rate=fs, 
    autoplay=True
)

plt.plot(eval(tune))
plt.yticks([])
plt.savefig('imgs/complex_example.png')

This is not going to sound very good, possibly because the grammar is not encoding any of the interesting structure we like in music.

Creating a noisy observation

tune = complete('<tune>', soundGrammar)
pprint(tune)
sound = eval(tune, interpret)
sigma = np.max(sound)/3
print(sigma)
noisy_sound = sound + np.random.normal(scale=sigma, size=sound.shape)

("melt(defineNote('A', 0.3), successive(melt(defineNote('G', 0.3), "
 "melt(successive(defineNote('A', 0.6), defineNote('A', 1.)), "
 "melt(successive(melt(defineNote('D', 0.6), successive(defineNote('C', 0.6), "
 "melt(successive(defineNote('C', 0.3), defineNote('D', 0.3)), defineNote('E', "
 "1.)))), defineNote('D', 0.6)), successive(defineNote('E', 0.6), "
 "defineNote('C', 1.))))), melt(successive(defineNote('E', 0.3), "
 "defineNote('C', 1.)), melt(defineNote('C', 1.), melt(defineNote('A', 0.6), "
 "defineNote('E', 0.3))))))")
10922.333333333334

print(tune)

successive(defineNote('G', 0.6), successive(defineNote('D', 0.6), successive(successive(successive(successive(defineNote('G', 0.3), melt(melt(defineNote('A', 0.3), successive(defineNote('C', 1.), defineNote('D', 0.3))), defineNote('G', 0.6))), defineNote('E', 0.6)), defineNote('E', 0.6)), successive(melt(melt(defineNote('E', 0.6), successive(defineNote('C', 0.3), defineNote('D', 1.))), defineNote('G', 0.3)), defineNote('G', 0.3)))))

IPython.display.Audio(
    # put note or notes here!
    sound, 
    rate=fs, 
    autoplay=True
)

IPython.display.Audio(
    # put note or notes here!
    noisy_sound, 
    rate=fs, 
    autoplay=True
)

plt.plot(sound)
plt.yticks([])
plt.savefig('imgs/clean_example.png')

plt.plot(noisy_sound)
plt.yticks([])
plt.savefig('imgs/noisy_example.png')

tune0 = complete('<tune>', soundGrammar)
tune1 = complete('<tune>', soundGrammar)
tune2 = complete('<tune>', soundGrammar)

print(tune0, '\n')
print(tune1, '\n')
print(tune2, '\n')

defineNote('C', 0.3) 

melt(defineNote('C', 0.3), successive(defineNote('G', 1.), defineNote('E', 0.6))) 

successive(melt(defineNote('C', 0.3), melt(successive(successive(defineNote('D', 0.3), defineNote('D', 1.)), defineNote('C', 0.3)), defineNote('E', 0.6))), defineNote('C', 0.3)) 

sound0 = eval(tune0, interpret)
sound1 = eval(tune1, interpret)
sound2 = eval(tune2, interpret)

sigma0 = np.max(sound0)/3
sigma1 = np.max(sound1)/3
sigma2 = np.max(sound2)/3

# make them all equally long
max_length = np.max([sound0.shape, sound1.shape, sound2.shape])

# diff from max length
sound0 = np.concatenate([
    sound0,
    np.zeros(np.abs(sound0.shape-max_length))
])

sound1 = np.concatenate([
    sound1,
    np.zeros(np.abs(sound1.shape-max_length))
])

sound2 = np.concatenate([
    sound2,
    np.zeros(np.abs(sound2.shape-max_length))
])

noisy_sound0 = sound0 + np.random.normal(scale=sigma0, size=sound0.shape)
noisy_sound1 = sound1 + np.random.normal(scale=sigma1, size=sound1.shape)
noisy_sound2 = sound2 + np.random.normal(scale=sigma2, size=sound2.shape)

fig, axes = plt.subplots(3,2,sharex=True)
((ax1, ax2, ax3), (noise_ax1, noise_ax2, noise_ax3)) = axes.T
ax1.plot(sound0)
noise_ax1.plot(noisy_sound0)
ax2.plot(sound1)
noise_ax2.plot(noisy_sound1)
ax3.plot(sound2)
noise_ax3.plot(noisy_sound2)

for ax in axes.flatten():
    ax.set_yticks([])

fig.tight_layout()
fig.savefig('imgs/prior_lik_compare.png')

IPython.display.Audio(
    # put note or notes here!
    sound0, 
    rate=fs, 
    autoplay=True
)

IPython.display.Audio(
    # put note or notes here!
    noisy_sound0, 
    rate=fs, 
    autoplay=True
)

IPython.display.Audio(
    # put note or notes here!
    sound1, 
    rate=fs, 
    autoplay=True
)

IPython.display.Audio(
    # put note or notes here!
    noisy_sound1, 
    rate=fs, 
    autoplay=True
)

IPython.display.Audio(
    # put note or notes here!
    sound2, 
    rate=fs, 
    autoplay=True
)

IPython.display.Audio(
    # put note or notes here!
    noisy_sound2, 
    rate=fs, 
    autoplay=True
)

Learning in an LoT

##### define a LOTlib3 grammar
lot_sound_grammar = Grammar(start='<tune>')
for nonterminal, substitutions in soundGrammar.items():
    for weight, subt in substitutions:
        to_substitute = []
        def capture_and_replace(match):
            to_substitute.append(match.group(0))
            return '%s'
        # replace nonterminals with %s
        subt_replaced = re.sub(
            r'<[a-z]*>', 
            capture_and_replace, 
            subt
        )
        # add a rule to the grammar
        print(subt_replaced)
        lot_sound_grammar.add_rule(
            nonterminal, 
            subt_replaced, 
            to_substitute if len(to_substitute)>0 else None, 
            weight
        )

print("Defined the grammar!")

%s
successive(%s, %s)
melt(%s, %s)
defineNote('%s', %s)
A
C
D
E
G
0.3
0.6
1.
Defined the grammar!

lot_sound_grammar.display_rules()

<tune> -> %s['<note>']	w/ p=3.0
<tune> -> successive(%s, %s)['<tune>', '<tune>']	w/ p=1.0
<tune> -> melt(%s, %s)['<tune>', '<tune>']	w/ p=1.0
<note> -> defineNote('%s', %s)['<freq>', '<time>']	w/ p=1.0
<freq> -> A	w/ p=1.0
<freq> -> C	w/ p=1.0
<freq> -> D	w/ p=1.0
<freq> -> E	w/ p=1.0
<freq> -> G	w/ p=1.0
<time> -> 0.3	w/ p=1.0
<time> -> 0.6	w/ p=1.0
<time> -> 1.	w/ p=1.0

def log_prob_normal(x, mu=0, sigma=1):
    """
    Returns the log-probability of x under a normal distribution
    with mean mu and standard deviation sigma.
    """
    return -0.5 * np.log(2 * np.pi) - np.log(sigma) - ((x - mu) ** 2) / (2 * sigma ** 2)

##### define a hypothesis
class MusicHypothesis(LOTHypothesis):
    
    def __init__(self, **kwargs):
        LOTHypothesis.__init__(
            self, 
            grammar=lot_sound_grammar, 
            display="lambda : %s", 
            **kwargs
        )
        self.sigma=sigma

    def compute_single_likelihood(self, datum):

        self_sound = self()
        obs_sound = datum.output

        # diff is positive when h is
        # longer than observation
        diff = len(self_sound) - len(obs_sound)
        if diff < 0:
            # self_sound = np.concatenate((
            #     self_sound,
            #     np.zeros(-diff, dtype=int)
            # ))
            # if observation is longer,
            # it can't be the right h
            return -np.inf
        elif diff > 0:
            obs_sound = np.concatenate((
                obs_sound,
                np.zeros(diff, dtype=int)
            ))
        logprob = log_prob_normal(
            self_sound - obs_sound,
            sigma=datum.sigma
        ).sum()
        
        return logprob

print("Defined a hypothesis class")

Defined a hypothesis class

tune = complete('<tune>', soundGrammar)
pprint(tune)

# evaluate the tune into soundwaves
sound = eval(tune, interpret)

# get a noise in the order of magnitude we want
sigma = np.max(sound)/1

noisy_sound = (
    sound 
    + np.random.normal(scale=sigma, size=sound.shape)
)

("melt(defineNote('G', 0.3), successive(successive(defineNote('A', 0.6), "
 "defineNote('C', 1.)), defineNote('E', 1.)))")

print(tune)

melt(defineNote('G', 0.3), successive(successive(defineNote('A', 0.6), defineNote('C', 1.)), defineNote('E', 1.)))

IPython.display.Audio(
    # put note or notes here!
    sound, 
    rate=fs, 
    autoplay=True
)

IPython.display.Audio(
    # put note or notes here!
    noisy_sound, 
    rate=fs, 
    autoplay=True
)

plt.plot(sound)
plt.yticks([])
plt.savefig('imgs/clean_example_learning.png')

plt.plot(noisy_sound)
plt.yticks([])
plt.savefig('imgs/noisy_example_learning.png')

##### Define Data
data = [ 
    FunctionData(
        input=[], 
        output=noisy_sound, 
        sigma=sigma
    )
]

# store the top N
tn = TopN(
    N=30
) 

##### Run inference
h = MusicHypothesis()

import sys
sys.setrecursionlimit(1500)

print("Running inference...")

for i, h in enumerate(MetropolisHastingsSampler(h, data, steps=10000)):
    tn.add(h)
    if i%100==0:
        print(i, end=', ')

Running inference...
0, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200, 1300, 1400, 1500, 1600, 1700, 1800, 1900, 2000, 2100, 2200, 2300, 2400, 2500, 2600, 2700, 2800, 2900, 3000, 3100, 3200, 3300, 3400, 3500, 3600, 3700, 3800, 3900, 4000, 4100, 4200, 4300, 4400, 4500, 4600, 4700, 4800, 4900, 5000, 5100, 5200, 5300, 5400, 5500, 5600, 5700, 5800, 5900, 6000, 6100, 6200, 6300, 6400, 6500, 6600, 6700, 6800, 6900, 7000, 7100, 7200, 7300, 7400, 7500, 7600, 7700, 7800, 7900, 8000, 8100, 8200, 8300, 8400, 8500, 8600, 8700, 8800, 8900, 9000, 9100, 9200, 9300, 9400, 9500, 9600, 9700, 9800, 9900, 

best_one = tn.best()

IPython.display.Audio(
    # put note or notes here!
    best_one(), 
    rate=fs, 
    autoplay=True
)

best_one

lambda : successive(successive(defineNote('A', 1.), defineNote('A', 1.)), defineNote('D', 0.6))

plt.plot(best_one())

[<matplotlib.lines.Line2D at 0x7fae69c4f3d0>]

IPython.display.Audio(
    # put note or notes here!
    sound, 
    rate=fs, 
    autoplay=True
)

for h in tn.get_all(sorted=True):
    print(np.exp(h.posterior_score), h.prior, h.likelihood, h)

0.0 -12.875503299472802 -1356873.3427921375 lambda : successive(successive(defineNote('G', 1.), defineNote('A', 1.)), defineNote('C', 0.6))
0.0 -12.875503299472802 -1356873.3383733607 lambda : successive(successive(defineNote('G', 1.), defineNote('A', 1.)), defineNote('G', 0.6))
0.0 -12.875503299472802 -1356873.3376659977 lambda : successive(successive(defineNote('G', 1.), defineNote('D', 1.)), defineNote('D', 0.6))
0.0 -12.875503299472802 -1356873.337366325 lambda : successive(successive(defineNote('C', 1.), defineNote('A', 1.)), defineNote('D', 0.6))
0.0 -12.875503299472802 -1356873.330359108 lambda : successive(successive(defineNote('G', 1.), defineNote('A', 1.)), defineNote('D', 0.6))
0.0 -12.875503299472802 -1356873.3051245434 lambda : successive(successive(defineNote('A', 1.), defineNote('C', 1.)), defineNote('A', 0.6))
0.0 -12.875503299472802 -1356873.2931516496 lambda : successive(successive(defineNote('A', 1.), defineNote('C', 1.)), defineNote('E', 0.6))
0.0 -12.875503299472802 -1356873.2907418034 lambda : successive(successive(defineNote('A', 1.), defineNote('C', 1.)), defineNote('C', 0.6))
0.0 -12.875503299472802 -1356873.2893205117 lambda : successive(successive(defineNote('A', 1.), defineNote('E', 1.)), defineNote('A', 0.6))
0.0 -12.875503299472802 -1356873.2863230265 lambda : successive(successive(defineNote('A', 1.), defineNote('C', 1.)), defineNote('G', 0.6))
0.0 -12.875503299472802 -1356873.282063569 lambda : successive(successive(defineNote('A', 1.), defineNote('G', 1.)), defineNote('A', 0.6))
0.0 -12.875503299472802 -1356873.278308774 lambda : successive(successive(defineNote('A', 1.), defineNote('C', 1.)), defineNote('D', 0.6))
0.0 -12.875503299472802 -1356873.2773476178 lambda : successive(successive(defineNote('A', 1.), defineNote('E', 1.)), defineNote('E', 0.6))
0.0 -12.875503299472802 -1356873.2752187664 lambda : successive(successive(defineNote('A', 1.), defineNote('D', 1.)), defineNote('A', 0.6))
0.0 -12.875503299472802 -1356873.2749377717 lambda : successive(successive(defineNote('A', 1.), defineNote('E', 1.)), defineNote('C', 0.6))
0.0 -12.875503299472802 -1356873.270518995 lambda : successive(successive(defineNote('A', 1.), defineNote('E', 1.)), defineNote('G', 0.6))
0.0 -12.875503299472802 -1356873.270090675 lambda : successive(successive(defineNote('A', 1.), defineNote('G', 1.)), defineNote('E', 0.6))
0.0 -12.875503299472802 -1356873.2679118768 lambda : successive(successive(defineNote('A', 1.), defineNote('A', 1.)), defineNote('A', 0.6))
0.0 -12.875503299472802 -1356873.267680829 lambda : successive(successive(defineNote('A', 1.), defineNote('G', 1.)), defineNote('C', 0.6))
0.0 -12.875503299472802 -1356873.263262052 lambda : successive(successive(defineNote('A', 1.), defineNote('G', 1.)), defineNote('G', 0.6))
0.0 -12.875503299472802 -1356873.2632458727 lambda : successive(successive(defineNote('A', 1.), defineNote('D', 1.)), defineNote('E', 0.6))
0.0 -12.875503299472802 -1356873.2625047422 lambda : successive(successive(defineNote('A', 1.), defineNote('E', 1.)), defineNote('D', 0.6))
0.0 -12.875503299472802 -1356873.2608360264 lambda : successive(successive(defineNote('A', 1.), defineNote('D', 1.)), defineNote('C', 0.6))
0.0 -12.875503299472802 -1356873.2564172498 lambda : successive(successive(defineNote('A', 1.), defineNote('D', 1.)), defineNote('G', 0.6))
0.0 -12.875503299472802 -1356873.255938983 lambda : successive(successive(defineNote('A', 1.), defineNote('A', 1.)), defineNote('E', 0.6))
0.0 -12.875503299472802 -1356873.2552477994 lambda : successive(successive(defineNote('A', 1.), defineNote('G', 1.)), defineNote('D', 0.6))
0.0 -12.875503299472802 -1356873.2535291368 lambda : successive(successive(defineNote('A', 1.), defineNote('A', 1.)), defineNote('C', 0.6))
0.0 -12.875503299472802 -1356873.24911036 lambda : successive(successive(defineNote('A', 1.), defineNote('A', 1.)), defineNote('G', 0.6))
0.0 -12.875503299472802 -1356873.2484029972 lambda : successive(successive(defineNote('A', 1.), defineNote('D', 1.)), defineNote('D', 0.6))
0.0 -12.875503299472802 -1356873.2410961075 lambda : successive(successive(defineNote('A', 1.), defineNote('A', 1.)), defineNote('D', 0.6))

hs = tn.get_all(sorted=True)
z = logsumexp([h.posterior_score for h in hs])
pps_sound = np.zeros(sound.shape)
for h in hs:
    p = np.exp(h.posterior_score - z)
    try:
        h_sound = h()
        print(p)
        print(h, "\n")
        pps_sound = pps_sound + h_sound*p
    except:
        print('Raised error')

0.03131755710576977
lambda : successive(successive(defineNote('G', 1.), defineNote('A', 1.)), defineNote('C', 0.6)) 

0.03145624859938691
lambda : successive(successive(defineNote('G', 1.), defineNote('A', 1.)), defineNote('G', 0.6)) 

0.031478507457739266
lambda : successive(successive(defineNote('G', 1.), defineNote('D', 1.)), defineNote('D', 0.6)) 

0.03148794211737613
lambda : successive(successive(defineNote('C', 1.), defineNote('A', 1.)), defineNote('D', 0.6)) 

0.03170935981671909
lambda : successive(successive(defineNote('G', 1.), defineNote('A', 1.)), defineNote('D', 0.6)) 

0.032519713163564244
lambda : successive(successive(defineNote('A', 1.), defineNote('C', 1.)), defineNote('A', 0.6)) 

0.03291140841703552
lambda : successive(successive(defineNote('A', 1.), defineNote('C', 1.)), defineNote('E', 0.6)) 

0.03299081549347328
lambda : successive(successive(defineNote('A', 1.), defineNote('C', 1.)), defineNote('C', 0.6)) 

0.033037738400539765
lambda : successive(successive(defineNote('A', 1.), defineNote('E', 1.)), defineNote('A', 0.6)) 

0.03313691710226045
lambda : successive(successive(defineNote('A', 1.), defineNote('C', 1.)), defineNote('G', 0.6)) 

0.033278363423400036
lambda : successive(successive(defineNote('A', 1.), defineNote('G', 1.)), defineNote('A', 0.6)) 

0.03340355173147385
lambda : successive(successive(defineNote('A', 1.), defineNote('C', 1.)), defineNote('D', 0.6)) 

0.03343567320104032
lambda : successive(successive(defineNote('A', 1.), defineNote('E', 1.)), defineNote('E', 0.6)) 

0.033506928598810944
lambda : successive(successive(defineNote('A', 1.), defineNote('D', 1.)), defineNote('A', 0.6)) 

0.0335163451907392
lambda : successive(successive(defineNote('A', 1.), defineNote('E', 1.)), defineNote('C', 0.6)) 

0.03366477413618318
lambda : successive(successive(defineNote('A', 1.), defineNote('E', 1.)), defineNote('G', 0.6)) 

0.03367919651764294
lambda : successive(successive(defineNote('A', 1.), defineNote('G', 1.)), defineNote('E', 0.6)) 

0.03375265668866999
lambda : successive(successive(defineNote('A', 1.), defineNote('A', 1.)), defineNote('A', 0.6)) 

0.03376045606872782
lambda : successive(successive(defineNote('A', 1.), defineNote('G', 1.)), defineNote('C', 0.6)) 

0.033909966072383405
lambda : successive(successive(defineNote('A', 1.), defineNote('G', 1.)), defineNote('G', 0.6)) 

0.03391051471977527
lambda : successive(successive(defineNote('A', 1.), defineNote('D', 1.)), defineNote('E', 0.6)) 

0.03393565614984416
lambda : successive(successive(defineNote('A', 1.), defineNote('E', 1.)), defineNote('D', 0.6)) 

0.03399233239224544
lambda : successive(successive(defineNote('A', 1.), defineNote('D', 1.)), defineNote('C', 0.6)) 

0.03414286926477765
lambda : successive(successive(defineNote('A', 1.), defineNote('D', 1.)), defineNote('G', 0.6)) 

0.034159202569025736
lambda : successive(successive(defineNote('A', 1.), defineNote('A', 1.)), defineNote('E', 0.6)) 

0.03418282101136813
lambda : successive(successive(defineNote('A', 1.), defineNote('G', 1.)), defineNote('D', 0.6)) 

0.034241620263676796
lambda : successive(successive(defineNote('A', 1.), defineNote('A', 1.)), defineNote('C', 0.6)) 

0.03439326112896527
lambda : successive(successive(defineNote('A', 1.), defineNote('A', 1.)), defineNote('G', 0.6)) 

0.03441759824829
lambda : successive(successive(defineNote('A', 1.), defineNote('D', 1.)), defineNote('D', 0.6)) 

0.03467000487070386
lambda : successive(successive(defineNote('A', 1.), defineNote('A', 1.)), defineNote('D', 0.6)) 

plt.plot(pps_sound)
plt.yticks([])
plt.savefig('imgs/pps_example_learning.png')

IPython.display.Audio(
    # put note or notes here!
    pps_sound, 
    rate=fs, 
    autoplay=True
)

tune

"melt(defineNote('A', 0.6), melt(melt(successive(defineNote('G', 0.6), melt(successive(defineNote('C', 0.6), defineNote('D', 1.)), successive(successive(defineNote('A', 0.3), defineNote('C', 0.3)), defineNote('C', 1.)))), melt(defineNote('G', 1.), defineNote('C', 1.))), successive(melt(defineNote('E', 0.3), defineNote('A', 1.)), defineNote('G', 0.6))))"

Things to try

• How could we change the vibe to be more calm?

• How could we modify the grammar to make tunes less ‘messy’?